On approximation to a real number by algebraic numbers of bounded degree

Abstract

In his seminal 1961 paper, Wirsing studied how well a given transcendental real number $ξ$ can be approximated by algebraic numbers $α$ of degree at most $n$ for a given positive integer $n$, in terms of the so-called naive height $H(α)$ of $α$. He showed that the infimum $ω^*_n(ξ)$ of all $ω$ for which infinitely many such $α$ have $|ξ-α| \le H(α)^{-ω-1}$ is at least $(n+1)/2$. He also asked if we could even have $ω^*_n(ξ) \ge n$ as it is generally expected. Since then, all improvements on Wirsing's lower bound were of the form $n/2+\mathcal{O}(1)$ until Badziahin and Schleischitz showed in 2021 that $ω^*_n(ξ) \ge an$ for each $n\ge 4$, with $a=1/\sqrt{3}\simeq 0.577$. In this paper, we use a different approach partly inspired by parametric geometry of numbers and show that $ω^*_n(ξ) \ge an$ for each $n\ge 2$, with $a=1/(2-\log 2)\simeq 0.765$.